Discrete-time Results
Similar results can be obtained for discrete-time supermartingales and submartingales, the obvious difference being that no continuity assumptions are required. For example, the result above becomes
Let M : N × Ω → R be a discrete-time martingale such that
for some p > 1. Then there exists a random variable M ∈ Lp(Ω, P; R) such that Mk → M as k → +∞ both P-almost surely and in Lp(Ω, P; R)
Read more about this topic: Doob's Martingale Convergence Theorems
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